Robin Coefficient Research Articles

Qualitative analysis of optimisation problems with respect to non-constant Robin coefficients

Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide in this article a detailed study of the optimisation of Robin boundary conditions in PDE constrained calculus of variations. Our main model consists of an elliptic PDE of the form −∆u β = f (x, u β) endowed with the Robin boundary conditions ∂ν u β +β(x)u β = 0. The optimisation variable is the function β, which is assumed to take values between 0 and 1 and to have a fixed integral. Two types of criteria are under consideration: the first one is non-energetic criteria. In other words, we aim at optimising functionals of the form J (β) =\int_{Ω or ∂Ω} j(u β). We prove that, depending on the monotonicity of the function j, the optimisers may be of bang-bang type (in other words, the optimisers write 1Γ for some measurable subset Γ of ∂Ω) or, on the contrary, that they may only take values strictly between 0 and 1. This has consequence for a related shape optimisation problem, in which one tries to find where on the boundary Neumann (∂ν u = 0) and constant Robin conditions (∂ν u + u = 0) should be placed in order to optimise criteria. The proofs for this first case rely on new fine oscillatory techniques, used in combination with optimality conditions. We then investigate the case of compliance-type functionals. For such energetic functionals, we give an in-depth analysis and even some explicit characterisation of optimal β * .

Convergence Rates for Identification of Robin Coefficient from Terminal Observations

This paper deals with the problem of identification of a Robin coefficient (also known as impedance coefficient) in a parabolic PDE from terminal observations of the temperature distributions. The problem is ill-posed in the sense that small perturbation in the observation may lead to a large deviation in the solution. Thus, in order to obtain stable approximations, we employ the Tikhonov-regularization. We propose a weak source condition motivated by the work of Engl and Zou (2000) and obtain a convergence rate of O(δ12)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\delta ^\\frac{1}{2})$$\\end{document}, where δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} is the noise level of the observed data. The obtained rate is better than some of the previous known rates. Moreover, the advantage of the proposed source condition is that the above mentioned convergence rate is obtained without the need for characterizing the range space of modelling operator and its Fréchet derivative, which is in contrast to the general convergence theory of Tikhonov-regularization for non linear operators.

Robin coefficient determination problem in a fractional parabolic differential equation

This article discusses the identification of an unknown Robin coefficient in a fractional parabolic equation using a noisy measurement of the ultimate solution over time. The problem is quite complex, involving a nonlocal, nonlinear, and ill‐posed operator. To solve this problem, the article proposes a regularized optimization approach that minimizes a least‐squares cost function. The article also examines the various conceptual and real‐world challenges associated with this problem. The article demonstrates the presence of a singular and stable solution for the optimization problem. It utilizes the Morozov discrepancy principle and the conjugate gradient method to streamline the iterative reconstruction process. The proposed method is accurate and efficient, as several numerical examples demonstrate.

Logarithmic type stability for the simultaneous identification of Robin coefficient and heat flux in an elliptic equation

In this paper, we aim to reconstruct the unknown Robin coefficient and unknown heat flux simultaneously in an elliptic system from Cauchy data on arbitrarily small portion of accessible boundary. A novel logarithmic type stability in terms of the negative Sobolev norms is established.

Multiple Terms Identification of Time Fractional Diffusion Equation with Symmetric Potential from Nonlocal Observation

This paper considers a simultaneous identification problem of a time-fractional diffusion equation with a symmetric potential, which aims to identify the fractional order, the potential function, and the Robin coefficient from a nonlocal observation. Firstly, the existence and uniqueness of the weak solution are established for the forward problem. Then, by the asymptotic behavior of the Mittag-Leffler function, the Laplace transform, and the analytic continuation theory, the uniqueness of the simultaneous identification problem is proved under some appropriate assumptions. Finally, the Levenberg–Marquardt method is employed to solve the simultaneous identification problem for finding stably approximate solutions of the fractional order, the potential function, and the Robin coefficient. Numerical experiments for three test cases are given to demonstrate the effectiveness of the presented inversion method.

Shape reconstruction for an inverse biharmonic problem from partial Cauchy data

In this paper, we address the Robin inverse problem for the biharmonic equation in a simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided.

Identify the Robin coefficient in an inhomogeneous time-fractional diffusion-wave equation

This paper is devoted to identifying the Robin coefficient in a one-dimensional inhomogeneous time-fractional diffusion-wave equation from the viewpoint of Bayesian inference approach. The priori modeling is implemented by a Markov Random Field (MRF). A hierarchical Bayesian model is adopted to select regularization parameters automatically. The Markov chain Monte Carlo (MCMC) algorithms are conducted to explore the posterior state space. Three numerical examples are carried out to demonstrate the performance of the proposed method.

A primal-dual approach for the Robin inverse problem in a nonlinear elliptic equation: The case of the 𝐿1 − 𝐿2 cost functional

Abstract In this work, we consider the inverse problem of identifying a Robin coefficient in a nonlinear elliptic equation with mixed boundary conditions. We firstly reformulate the inverse problem as a regularized optimal control one, where the functional cost is of type L 1 - L 2 L^{1}-L^{2} ; then we prove the existence and uniqueness of a minimizer to the resulting optimization problem in a suitable functional space. Finally, we provide a primal-dual algorithm to solve the variational problem and give some numerical results that prove the accuracy of the proposed method in the identification of the Robin coefficient.

Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data.

Second-order diffusion limit for the phonon transport equation: asymptotics and numerics

We investigate the numerical implementation of the limiting equation for the phonon transport equation in the small Knudsen number regime. The main contribution is that we derive the limiting equation that achieves the second order convergence, and provide a numerical recipe for computing the Robin coefficients. These coefficients are obtained by solving an auxiliary half-space equation. Numerically the half-space equation is solved by a spectral method that relies on the even-odd decomposition to eliminate corner-point singularity. Numerical evidences will be presented to justify the second order asymptotic convergence rate.

Simultaneous recovery of Robin boundary and coefficient for the Laplace equation by shape derivative

We study the simultaneous recovery of the boundary and coefficient of the Robin boundary condition for the Laplace equation from a pair of solution measurements on another part of the boundary. We derive the variational derivatives of the data-fitting objective functional with respect to the Robin boundary and coefficient, which are then used to device a nonlinear conjugate gradient iterative scheme for the numerical recovery of both the Robin boundary and coefficient together. Numerical examples are presented to illustrate the effectiveness of the recovery algorithms.

Retrieving the Robin coefficient from single Cauchy data in elliptic systems

The purpose of this work is to identify a Robin coefficient from available measurements on the accessible part of the boundary. After recasting the inverse problem as an optimization problem, we study the issue of identifiability, stability, and identification. For the reconstruction process, two regularized algorithms are designed, and the forward problem is approximated using the discontinuous dual reciprocity method. The accuracy of the proposed approaches is tested in the case of noise–free and noisy data and the findings are very promising and encouraging.

Joint estimation of Robin coefficient and domain boundary for the Poisson problem

We consider the problem of simultaneously inferring the heterogeneous coefficient field for a Robin boundary condition on an inaccessible part of the boundary along with the shape of the boundary for the Poisson problem. Such a problem arises in, for example, corrosion detection, and thermal parameter estimation. We carry out both linearised uncertainty quantification, based on a local Gaussian approximation, and full exploration of the joint posterior using Markov chain Monte Carlo sampling. By exploiting a known invariance property of the Poisson problem, we are able to circumvent the need to re-mesh as the shape of the boundary changes. The linearised uncertainty analysis presented here relies on a local linearisation of the parameter-to-observable map, with respect to both the Robin coefficient and the boundary shape, evaluated at the maximum a posteriori (MAP) estimates. Computation of the MAP estimate is carried out using the Gauss–Newton method. On the other hand, to explore the full joint posterior we use the Metropolis-adjusted Langevin algorithm, which requires the gradient of the log-posterior. We thus derive both the Fréchet derivative of the solution to the Poisson problem with respect to the Robin coefficient and the boundary shape, and the gradient of the log-posterior, which is efficiently computed using the so-called adjoint approach. The performance of the approach is demonstrated via several numerical experiments with simulated data.

On the simultaneous reconstruction of boundary Robin coefficient and internal source in a slow diffusion system

Consider an inverse problem of simultaneous reconstruction of boundary impedance coefficient and space-dependent source term from the final measurement data for a slow diffusion system, which is governed by a diffusion equation with time-fractional order derivative and Robin boundary condition. We firstly prove the uniqueness of this inverse problem by the maximum principle for the slow diffusion system. Then a regularizing scheme combining the mollification method and the Tikhonov regularization is proposed to recover the two unknowns, with a rigorous analysis on the choice strategies for the regularizing parameters and the error estimates on the regularizing solutions, revealing the error propagation effects due to recovering the boundary Robin coefficient firstly. Numerical examples are presented to illustrate the validity of the proposed method and support our theoretical analysis.

A reduced basis Landweber method for the identification of piecewise constant Robin coefficient in an elliptic equation

Abstract In this paper, we are concerned with the identification of the piecewise constant Robin coefficient in an elliptic equation. The iterative regularization method is one of the very effective methods for solving this kind of nonlinear ill-posed inverse problems. But it usually requires to solve numerous amounts of forward solutions during the iterative process, which will cost a lot of computational time in high-dimensional spaces. A reduced basis method is considered to reduce the computational time for solving the forward problems, and its error estimate is also studied. Finally, we propose a reduced basis Landweber algorithm to solve the elliptic inverse Robin problem and present several numerical experiments to demonstrate the accuracy and efficiency of the algorithm.

Integral equations method for solving a Biharmonic inverse problem in detection of Robin coefficients

In this paper, we are interested in an inverse problem for the biharmonic equation to recover the Robin coefficients on a non-accessible part of the boundary in simply connected planar domain from a measured Riquier-Neumann data on the accessible part of the boundary. Our work is based on a system of nonlinear and ill-posed integral equations which is solved by Tikhonov regularization method to complete missing Cauchy data. The minimization problem will be considered to solve the inverse problem, and recover the Robin coefficients on a known boundary curve. We show the feasibility of this method by numerical reconstructions.

Inexact Newton regularization in Banach spaces based on two-point gradient method with uniformly convex penalty terms

In this paper, we propose and analyze a new inexact Newton regularization method for solving nonlinear inverse problems in Banach spaces. The method consists of an outer Newton iteration and an inner scheme which provides increments by applying a regularization technique to the local linearized equation around the current iterate. In order to accelerate the convergence, we employ a two-point gradient method as inner regularization scheme, which is based on the Landweber iteration and an extrapolation strategy. The penalty term is allowed to be non-smooth, including L1-like and total variation-like penalty functionals, to detect special features of solutions such as sparsity and piecewise constancy. We present, under certain assumptions, the detailed analysis of convergence and regularization properties of the method. Finally, some numerical experiments on elliptic parameter identification and Robin coefficient reconstruction problems are provided to illustrate the effectiveness of reconstructing the properties of sought solutions and the acceleration effect of the method.

Identification of piecewise constant Robin coefficient for the Stokes problem using the Levenberg–Marquardt method

In this work, we prove the quadratic convergence of the Levenberg–Marquardt method for the inverse problem of identifying a Robin coefficient for the Stokes system, where we suppose that this parameter is piecewise constant on some non-accessible part of the boundary and under the assumption that on this part, the velocity of a given reference solution stays far from zero.

On an inverse Robin spectral problem

We consider the problem of the recovery of a Robin coefficient on a part γ ⊂ ∂Ω of the boundary of a bounded domain Ω from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on ∂Ω\\γ. We prove the uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present an iterative reconstruction algorithm with numerical computations in two dimensions showing the accuracy of the method.

Quantum vacuum under mixed boundaryconditions: the case for curved spacetime

Influence of gravity on the quantum vacuum of a massless minimally coupled scalar field under Robin boundary conditions on parallel plates is investigated. We introduce the detailed calculation of the volume energy for the case the gravitational background is weak in its most general form for a static spacetime. It founds that the quantum vacuum usually reacts to the gravitational field by decreasing the Casimir energy. In addition, we find sufficient conditions under which the Casimir force increases. Interestingly, the first order perturbation corrections, are present in the obtained formula for the volume energy. We show that for some specific choices of parameters, the energy is independent of Robin coefficients. Consistency with the literature is shown in some limiting cases and well-known examples are presented for both an increase or decrease in the volume energy.